Chapter 6 — The Triangle And Its Properties

Triangle — definition: A triangle is a polygon with three sides and three vertices. It is named by its vertices, e.g., triangle ABC has vertices A, B and C and sides AB, BC and CA.

Parts of a triangle:

• Vertices: The three corner points (A, B, C).
• Sides: Line segments joining the vertices (AB, BC, CA).
• Interior angles: Angles at the vertices (∠A, ∠B, ∠C).
• Base: Any side chosen as the base (often BC is taken as base).
• Altitude or height: The perpendicular from a vertex to the opposite side (or its extension).
• Median: A line segment joining a vertex to the midpoint of the opposite side; it divides the side into two equal parts.
• Angle bisector: A line that divides an angle into two equal angles.
• Perpendicular bisector: A line perpendicular to a side at its midpoint; points on it are equidistant from the side's endpoints.
• Exterior angle: An angle formed when a side of a triangle is extended; each exterior angle equals the sum of the two opposite interior angles.

Classification by sides:

• Equilateral: All three sides equal; all angles 60°.
• Isosceles: Two sides equal; base angles are equal.
• Scalene: All three sides are different.

Classification by angles:

• Acute triangle: All three angles are less than 90°.
• Right triangle: One angle is 90°; the side opposite the right angle is the hypotenuse.
• Obtuse triangle: One angle is greater than 90°.

Key properties:

• The sum of the three interior angles of a triangle is 180°: ∠A + ∠B + ∠C = 180°.
• An exterior angle equals the sum of the two opposite interior angles: exterior at A = ∠B + ∠C.
• Triangle inequality: The sum of any two sides is greater than the third side: AB + BC > CA, BC + CA > AB, CA + AB > BC.
• Perimeter = sum of all three sides.
• Area (basic formula): area = (1/2) × base × height.

These definitions and properties form the foundation for understanding construction, measurement and congruence of triangles in later topics.

A triangle is a three-sided polygon. Triangles are classified in two common ways: by their sides and by their angles.

Classification by sides

• Equilateral triangle: All three sides are equal. All three interior angles are equal (each 60°).
• Isosceles triangle: Two sides are equal. The angles opposite the equal sides (base angles) are equal.
• Scalene triangle: All three sides are of different lengths and all three angles are different.

Classification by angles

• Acute triangle: All three interior angles are less than 90°.
• Right triangle: One interior angle is exactly 90° (a right angle).
• Obtuse triangle: One interior angle is greater than 90° (an obtuse angle).

Important properties (Class 7 level)

• The sum of the interior angles of any triangle is 180°.
• An exterior angle of a triangle equals the sum of the two opposite interior angles.
• In an isosceles triangle, the base angles are equal.
• In an equilateral triangle, all sides and all angles are equal (each angle = 60°).
• Triangle inequality: the sum of the lengths of any two sides is always greater than the third side.
• Perimeter of a triangle = sum of its three side lengths. Area (basic formula) = 1/2 × base × height.

How to identify quickly

• Look at side lengths (or mark equal-looking sides) to decide equilateral / isosceles / scalene.
• Estimate angles; if you see a right-angle square marker, it is a right triangle. If one angle looks bigger than a right angle it is obtuse; if all look acute it is acute.

Statement: The sum of the three interior angles of any triangle is 180°.

Why this is true (simple proof using a parallel line):

1. Take triangle ABC. We want to show ∠A + ∠B + ∠C = 180°.
2. Through vertex C draw a line parallel to side AB (call it l). Since l || AB, the angle formed at C by this line and AC is equal to ∠A (they are alternate interior angles). Similarly, the angle formed at C by this line and BC equals ∠B.
3. Those two angles at C together with ∠C lie on a straight line l and therefore add up to 180° (a straight angle). So ∠A + ∠B + ∠C = 180°.

Key consequences (corollaries):

• Each interior angle of a triangle is less than 180° and, in fact, less than 180° by the sum of the other two.
• An exterior angle of a triangle equals the sum of the two opposite interior angles (Exterior Angle Theorem).
• The sum of the three exterior angles (one at each vertex) taken in the same orientation is 360°.

Class 7 level remark: This property holds for all triangles (scalene, isosceles, equilateral, acute, obtuse, right) and is used to find a missing angle when the other two are known.

Definition: An exterior angle of a triangle is formed when one side of the triangle is produced (extended) beyond a vertex. For example, in triangle ABC if side BC is extended to D, the angle ACD (outside the triangle) is an exterior angle at C.

Exterior angle property (Class 7 level): The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior (non-adjacent) angles.

Simple proof (using angle sum of triangle):

• In triangle ABC, extend BC to D to form exterior angle ACD.
• Interior angles of triangle: ∠A + ∠B + ∠C = 180°.
• Angles on a straight line: ∠ACD + ∠C = 180° (linear pair).
• From the two equations, ∠ACD = 180° − ∠C = (∠A + ∠B + ∠C) − ∠C = ∠A + ∠B.
• Hence exterior angle ∠ACD = ∠A + ∠B (sum of two opposite interior angles).

Corollaries:

• An exterior angle of a triangle is greater than either of the two opposite interior angles (since it equals their sum).
• This property helps find unknown angles quickly when one exterior and one interior (or two interiors) are known.

Why it matters: This property is a basic and frequently used result in geometry—useful in solving angle-chasing problems, proving facts about polygons, and in real-life applications like construction and design.

Definitions

An isosceles triangle has two equal sides (called the legs) and a third side called the base. An equilateral triangle has all three sides equal.

Key properties of an isosceles triangle (say triangle ABC with AB = AC)

• Base angles are equal: ∠ABC = ∠ACB (base-angle theorem).
• The altitude from the vertex A to the base BC also acts as the median and angle bisector: it is perpendicular to BC, bisects BC, and bisects ∠A. In other words, the perpendicular from A to BC meets BC at its midpoint.
• If two angles of a triangle are equal, the sides opposite them are equal (converse): equality of angles implies the triangle is isosceles.
• There is exactly one line of symmetry (the axis through the vertex and midpoint of the base).

Key properties of an equilateral triangle (side length a)

• All sides equal and all interior angles are 60°.
• Altitude, median, perpendicular bisector and angle bisector from any vertex coincide.
• Centroid, incenter, circumcenter and orthocenter all coincide at the same point.
• Three lines of symmetry (one through each vertex and opposite side midpoint) and rotational symmetry of order 3.

Why these hold (brief idea)

The equal-side → equal-angle result is proved by splitting the isosceles triangle along the altitude from the apex. This creates two congruent right triangles (by SSS or RHS), from which equal angles and equal segments follow. For equilateral triangles the same argument applied to any side shows all such special lines coincide because all three sides and three angles are identical.

Basic notation: In triangle ABC, side a = BC (opposite ∠A), b = CA (opposite ∠B), c = AB (opposite ∠C).

Main ideas:

• Equal sides ⇄ equal opposite angles: If two sides of a triangle are equal, then the angles opposite those sides are equal. (Isosceles triangle property.)
• Greater side ⇄ greater opposite angle: In any triangle, the greater side is opposite the greater angle, and conversely the greater angle is opposite the greater side.

Why these are true (sketches of proofs):

1. Equal sides ⇒ equal opposite angles: If AB = AC in triangle ABC, compare triangle ABC with itself by swapping B and C. Using side-side-side (or simple symmetry), the triangles are congruent so ∠B = ∠C.
2. Greater side ⇒ greater opposite angle (proof idea): Suppose AB > AC in triangle ABC. Mark point D on AB so that AD = AC. Triangle ADC is isosceles, so ∠ACD = ∠ADC. But ∠ABC (the whole angle at B) is larger than ∠ADC (part of it). Therefore ∠ABC > ∠ACB. So the angle opposite the longer side AB is larger than the angle opposite the shorter side AC.
3. Converse (greater angle ⇒ greater side): If ∠B > ∠C then the side opposite ∠B (which is b = AC) is longer than the side opposite ∠C (c = AB). A proof uses contradiction: if AC ≤ AB, previous result would give ∠B ≤ ∠C, contradiction; hence AC > AB.

Important remarks:

• All comparisons assume a non-degenerate triangle (three non-collinear points).
• These relations let you compare unknown sides or angles: if you can compare two angles you can compare the opposite sides, and vice versa.

Definition: In any triangle, the sum of the lengths of any two sides is always greater than the length of the third side. If the sides are a, b and c, then

a + b > c,   b + c > a,   c + a > b.

Why it is true (intuitive proof): The straight line is the shortest distance between two points. To go from vertex A to vertex C directly you take side AC. If you go from A to C via B you travel AB + BC, which must be longer than AC. The same reasoning applies to each pair of sides. Thus each side is less than the sum of the other two.

Important note: If the sum of two sides equals the third (for example a + b = c), the three points lie on a straight line and do not form a triangle (this is called a degenerate triangle). For a valid (non-degenerate) triangle the inequalities are strict (>).

What is a converse statement?
Every theorem has the form "If P then Q." The converse swaps hypothesis and conclusion: "If Q then P." A converse may be true or false; if true it is often proved separately.

What is a corollary?
A corollary is a result that follows easily from a theorem (or from a theorem plus a little extra reasoning).

Common theorems, their converses and corollaries in triangles (Class 7 level)

• 1. Base-Angles Theorem (Isosceles triangle theorem)
  Theorem: In triangle ABC, if AB = AC (two sides equal) then ∠B = ∠C (base angles are equal).
  Converse: If ∠B = ∠C then AB = AC (if two base angles are equal the opposite sides are equal — the triangle is isosceles).
  Corollary: If two sides are equal then the altitude/median/angle-bisector from the vertex between those sides are the same line (in an isosceles triangle the altitude from apex to base is also the median and the angle-bisector).
• 2. Midpoint (Mid-segment) Theorem
  Theorem: If D and E are midpoints of AB and AC in triangle ABC, then DE ∥ BC and DE = (1/2)·BC.
  Converse (useful form): If a line through the midpoint of one side of a triangle is parallel to a second side, it meets the third side at its midpoint. (So: a line through midpoint of AB parallel to BC meets AC at its midpoint.)
  Corollary: The segment joining midpoints of two sides divides the triangle into two smaller triangles of equal area.
• 3. Exterior-Angle Remark and corollary
  Theorem (exterior angle): An exterior angle of a triangle equals the sum of the two opposite interior angles and is greater than either of them.
  Corollaries: (a) Any exterior angle > any one opposite interior angle. (b) If an angle is equal to the sum of two other angles, it can be an exterior angle formed by extending the appropriate side.
• 4. Relationship between equal angles and symmetrical properties
  Corollary from base-angles theorem: If a triangle has two equal angles, it is symmetric about the perpendicular from the apex to the base; construction and measurement properties follow (useful in compass-and-straightedge constructions).

How to use converse and corollary in problem solving
- To prove a side is equal, you can try to prove equal opposite angles and then use the converse of the base-angles theorem.
- If you know a segment is parallel and one endpoint is a midpoint, use the midpoint converse to get the other midpoint.
- Use corollaries to get quick facts (equal areas, combined equalities) without re-proving the full theorem.

Note on logic: Always check whether a converse is valid before using it. If it is not stated as a true theorem, provide a short proof or counterexample.

Overview: This topic deals with solving problems about triangles, proving fundamental properties, and seeing how those properties are useful in real life. Key properties include the sum of interior angles, the exterior-angle theorem, the triangle inequality, and special facts about isosceles and scalene triangles. Understanding proofs helps you apply these facts correctly in constructions and problem solving.

1. Sum of interior angles = 180° (Proof)
Let triangle ABC be given. Through vertex C draw a line parallel to AB. The angle at A and the angle formed at C with that parallel line are alternate interior angles, so they are equal. Similarly the angle at B equals the other angle formed at C. Thus the three interior angles of triangle ABC form a straight line at C, so their sum = 180°.

2. Exterior-angle theorem (Proof)
If at vertex B of triangle ABC we extend side BC to D, then angle ABD (the exterior angle at B) equals the sum of the two opposite interior angles (angle A + angle C). Proof: Using the straight-line relation at B, angle ABD = 180° − angle ABC. From (1), angle A + angle B + angle C = 180°, so angle A + angle C = 180° − angle B = angle ABD.

3. Triangle inequality (Proof)
In any triangle, the sum of lengths of any two sides is greater than the third side. For example, AB + BC > AC. Proof (geometric idea): Place triangle ABC with A and B along a line and extend AB past B to a point D so that BD = BC. Then AC is shorter than AD = AB + BD = AB + BC, so AB + BC > AC. This works for any choice of two sides.

4. Isosceles triangle property (Proof)
If two sides of a triangle are equal (say AB = AC), then base angles are equal (angle B = angle C). Proof idea: Draw perpendicular/ or use triangle congruence: triangles formed by dropping a perpendicular from A to BC or by considering triangles ABC and ACB show corresponding sides and angles equal.

Common problem types include: proving equal angles or sides, computing a missing angle using angle-sum or exterior-angle theorem, checking inequalities for side lengths, and constructing triangles with given data (side-side-side, side-angle-side, etc., where permissible).

Applications: Designing roof trusses (isosceles shapes), measuring heights/ distances indirectly, checking structural stability (triangle inequalities ensure members reach), navigation and surveying (triangulation uses triangle angle relations), and constructing rigid frames (triangles provide rigidity).

How to approach problems:

1. Draw a clear diagram and label all known quantities.
2. Look for parallel lines, straight lines, or equal sides/angles to apply theorems.
3. Use the angle-sum (180°) and exterior-angle theorem to find missing angles.
4. Use the triangle inequality when comparing side lengths or deciding feasibility of construction.
5. When asked to prove, state given information, what to prove, and give a sequence of logical steps referring to earlier theorems.

Tips for proofs: Use parallel-line arguments (alternate interior/ corresponding angles), extend sides to form exterior angles, and use congruence of smaller triangles where appropriate.